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Kant and real numbersMark van Atten

To cite this version:Mark van Atten. Kant and real numbers. Peter Dybjer, Sten Lindström, Erik Palmgren, GöranSundholm. Epistemology versus Ontology: Essays on the Philosophy and Foundations of Mathematicsin Honour of Per Martin-Löf, 27, Springer, pp.203-213, 2012, Logic, Epistemology, and the Unity ofScience, 978-94-007-4434-9. �10.1007/978-94-007-4435-6_10�. �halshs-00775352�

https://halshs.archives-ouvertes.fr/halshs-00775352http://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/https://hal.archives-ouvertes.fr

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Kant and real numbers

Mark van AttenSND (CNRS / Paris IV), 1 rue Victor Cousin, 75005 Paris, France.

vanattenmark@gmail.com

dedicated to Per Martin-Löf

Kant held that under the concept of √2 falls a geometrical magnitude,but not a number. In particular, he explicitly distinguished this root frompotentially infinite converging sequences of rationals. LikeKant, Brouwerbased his foundations of mathematics on the a priori intuition of time,but unlike Kant, Brouwer did identify this root with a potentially infinitesequence. In this paper I discuss the systematical reasons why in Kant’sphilosophy this identification is impossible.

1 Introduction

Consider the following three concepts:

1. The square root of 2

2. The diagonal of a square with sides of length 1

3. The infinite sequence of rational numbers

1, 1.4, 1.41, . . .

given by a rule that ensures that the square of the successive rationalsconverges to 2.

1

Nowadays we say that under each of these concepts falls an object. The lengthof the diagonal of the square is given by √2 but this root exists independentlyfrom geometry.√2 is a real number, and this real number can, if one wishes todo so, be identified with the infinite sequence determined by the third concept(or, alternatively, an equivalence class of such sequences).

But Kant stopped short, like a horse in front of a fence, of introducing realnumbers by identifying them with infinite sequences. Indeed, he viewed therelations between the three concepts above differently. The main source thatdocuments Kant’s view on real numbers is his reply of Autumn 1790 to a letterfrom August Rehberg.1 This view was, as such, perfectly traditional in Kant’sdays,2 but it is interesting to see that they readily fit his newly proposed founda-tions of mathematics:3

1. Rehberg’s letter: AA XI:205–206; Kant’s reply: AA XI:207–210. Rehberg did not reply inturn, but much later published excerpts from Kant’s letter to him, together with his dissat-isfied comments on it, in the first volume of his Saemtliche Schriften of 1828 (Rehberg 1828,pp. 52–60). With the publication of Kant’s Nachlass, also two drafts for his reply to Rehbergbecame known (AAXIV:53–55,55–59). For an amusing description ofRehberg, see Jachmann’sletter to Kant of October 14, 1790, AA XI:215–227, in particular p. 225. It is Jachmann’s letterthat tells us that Rehberg’s letter came to Kant via Nicolovius. For detailed information onRehberg’s life and work, see Beiser 2008.

2. However, Stevin had already argued in 1585 that√8 is a number because it is part of 8, whichis a number: ‘La partie est de la mesme matiere qu’est son entier ; Racine de 8 est partie de sonquarré 8 : Doncques √8 est de la mesme matiere qu’est 8 : Mais la matiere de 8 est nombre ;Doncques lamatiere de√8 est nombre : Et par consequent√8, est nombre.’ Of course, Stevindid not go on to provide an arithmetization of real numbers (Stevin 1585, p. 30).

3. Kant didnot publish this view inhis lifetime, and it seems it first appeared inprint inRehberg’slater comments on their exchange (Rehberg 1828, pp. 52–60). However, three remarks to thesame effect were published within a framework close to Kant’s in Solomon Maimon’s bookon Kant’s philosophy, Versuch über die Transscendentalphilosophie of Autumn 1789, the yearbeforeKant’s exchangewithRehberg (1790; the title page states 1790, but see its editor’s remarkin footnote 1 on p. II of the edition used here). The remarks in question appear on p. 374,229/374, and 374, respectively. There seems to be no evidence as to whether Kant had seenMaimon’s remarks before writing to Rehberg (or later). (Warda’s list (Warda 1922) and themore comprehensive database ‘Kants Lektüre’ (URL = http://web.uni-marburg.de/kant/webseitn/ka_lektu.htm) suggest that Kant did not own Maimon’s book. But that does notshow that he did not see it at some point.) Note that Rehberg, in his later comments (Rehberg

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1. The concept of the magnitude√2 is not empty, because it can be instan-tiated geometrically:

That a middle proportional magnitude can now be found betweenone that equals 1 and another that equals 2, and is therefore notan empty concept (without an object), geometry shows with thediagonal of the square. (AA XI:208)4

2. But√2 cannot be instantiated numerically, because genuine numbers arecomposed out of units and hence rational:

But the pure schema of magnitude (quantitatis), as a concept of theunderstanding, is number, a representation which comprises thesuccessive addition of homogeneous units. (A142/B182, also referredto by Rehberg)5

So the only question is why for this quantum [√2] no number canbe found that represents the quantity (the ratio to unity) clearlyand completely in a concept. … That, however, the understanding,which arbitrarily makes for itself the concept of √2, cannot alsobring forth the complete number concept, namely its rational ratioto unity, … (AA XI:208)6

1828), does not mention Maimon’s book either. We will come back to the exchange betweenRehberg and Kant from a systematical point of view in Sect. 3. Note added in 2017: Thatdatabase can now be found at http://www.online.uni-marburg.de/kant_old/webseitn/ka_lektu.htm.

4. ‘Daßnundiemittlere Proportionalgröße zwischen einer die= 1und einer anderenwelche= 2gefunden werden könne, mithin jene kein leerer Begrif (ohne Object) sey, zeigt die Geometriean der Diagonale des Qvadrats.’

5. ‘Das reine Schema der Größe aber (quantitatis), als eines Begriffs des Verstandes, ist die Zahl,welche eine Vorstellung ist, die die sukzessive Addition von Einem zu Einem (gleichartigen)zusammenbefaßt.’

6. ‘Es ist also nur die Frage warum für diesesQvantum [√2] keine Zahl gefundenwerden könne

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3. Wemay have a rule to generate a potentially infinite sequence of rationalsthat will approximate an irrational ‘number’ such as√2:

that for every number one should be able to find a square root, ifnecessary one that is itself no number, but only the rule to approxi-mate it as closely as one wishes, … (AA XI:210)7

But for Kant √2 and a sequence of rational approximations to it are twodifferent things. This becomes clear when in his reply to Rehberg he writes ofsuch a sequence as

a sequence of fractions that, because it can never be completed, althoughit can be brought as near to completion as one wishes, expresses the root(but only in an irrational way) (AA XI:209)8

Indeed, this incompletability served to characterize ‘irrational’ in one of Kant’sReflexionen of the same period:

Concepts of irrational ratios are those, that cannot be exhausted by anyapproximation. (AA XVIII:716, 1790–1795?)9

If Kant had thought that the square root could be identified with the poten-tially infinite sequence, he could not have said that the latter is an incomplete

welche die Qvantität (ihr Verhaltnis zur Einheit) deutlich und vollständig im Begriffe vor-stellt. … Daß aber der Verstand, der sich willkürlich den Begrif von√2√2macht, nicht auchden vollständigen Zahlbegrif, nämlich durch das rationale Verhaltnis derselben zur Einheithervorbringen könne, … ’

7. ‘daß sich zu jeder Zahl eineQvadratwurzel finden lassenmüsse, allenfalls eine solche, die selbstkeine Zahl, sondern nur die Regel der Annäherung derselben, wie weit man es verlangt, … ’

8. ‘eine…Reihe vonBrüchen…, die, weil sie nie vollendet seyn kan, obgleich sich derVollendungso nahe bringen läßt als man will, die Wurzel (aber nur auf irrationale Art) ausdrückt’.

9. ‘Begriffe irrationaler Verhaltnisse sind solche, die durch keine Annäherung erschopft werdenkönnen.’

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expression of the former.10 Thus, although Kant at times speaks of ‘irrationalnumbers’,11 he alsomade it clear that this is a façon de parler, and that we actuallyonly have rules for generating approximations.

In a foundational setting, to introduce real numbers as infinite sequences, onehas to do two things:

1. Give a foundational account of infinite sequences as objects;

2. Explain in what sense such sequences can be considered to be numbers.

So when Kant rejects the identification, that can be on account of his conceptof number, or on account of his foundational ideas about infinite sequences.Indeed, in his writings and correspondence, one finds objections of both kinds.

Kant’s conception,mentioned above, that a number is composed out of givenunits, and that accordingly onlywhole numbers and rationals are numbers in theproper sense, goes back to the Greek.12 Note that Kant never adopted the moregeneral concept of number as proportion, as, for example, Newton had;13 for the

10. Note that, in the same sense, an infinite decimal expansion such as 0.333 . . . would also be‘an irrational way’ to express a magnitude, but in that case there is also the rational way ofexpressing it as a complete object, i.e., the fraction 1/3. Hegel called the expression by infiniteand hence incompletable means of something that can also be expressed finitely and hencecompletely a case of ‘bad infinity’ (‘schlechte Unendlichkeit’) (his example being the infinitedecimal expansion 0.285714 . . . and the fraction 2/7) (Hegel 1979, pp. 287–289). I thankPirmin Stekeler-Weithofer for bringing this to my attention.

11. A480/B508, AA XI:209, AA XIV:57, AA XVII:718.12. For example, both Euclid (Elements, Book VII, def. 2) and Diophantus (Arithmetic, Book

I, Introduction) define numbers as multitudes of units; while Euclid did not accept rationalnumbers, Diophantus did, in the sense that, as Klein explains it, ‘by a fraction Diophantusmeant nothing but a number of fractional parts’ (Klein 1968, p. 137).

13. ‘By number we understand not a multitude of units, but rather the abstract ratio of anyone quantity to another of the same kind taken as unit. Numbers are of three sorts; integers,fractions, and surds: an integer is what the unit measures, the fraction what a submultiplepart of the unit measures, and a surd is that with which the unit is incommensurable.’ (‘Pernumerum non tam multitudinem unitatum quam abstractam quantitatis cujusvis ad aliamejusdem generis quantitatem quae pro unitate habetur rationem intelligimus. Estque triplex;integer, fractus & surdus: Integer quem unitas metitur, fractus quem unitatis pars submulti-plex metitur, & surdus cui unitas est incommensurabilis.’) Newton,Arithmetica universalis,

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question addressed in the present paper, it would have made little difference ifhe had.14

Today, on the other hand, one would defend the claim that certain infinitesequences can be said to be numbers by referring to the algebraic concept of afield extension of the rationals. That is the result of a development starting withAbel and Galois and in whichHankel’sTheorie der complexen Zahlensysteme of1867 was of particular importance.15 Note that this algebraic concept is abstractenough not to depend on the particular way real numbers are implemented.Indeed, while Hankel thus extended the traditional concept of number, he stillheld that the existence of irrational numbers is shownonly geometrically (Hankel1867, p. 59);16 his work preceeded the arithmetization of real numbers as infinitesequences of rationals by Cantor in 1872 (Cantor 1872).17 Cantor comfortablycounted these sequences among the ‘numerical quantities’ (‘Zahlengrösse’),18and emphasized that on this conception, the real number is not an object that isdistinct from the sequence, as limits in the original sense were.

Interestingly, Charles Méray, who three years before Cantor published thesame mathematical ideas (Méray 1869), and thus holds priority,19 had still pre-

sive de compositione et resolutione arithmetica liber (Cambridge, 1707) as quoted and trans-lated in Petri and Schappacher 2007, p. 344.

14. Eudoxus’ theory of proportions was, however, of great importance to Kant’s views on therelations between arithmetic, geometry, and algebra. For an extensive treatment of that topic,see Sutherland 2006.

15. I thank Carl Posy for drawing my attention to this.16. Tennant 2010, “Why arithmetize the reals? Why not geometrize them?” (unpublished type-

script) addresses the following question: ‘Who was the first major foundationalist thinkerexplicitly to reject (on the basis of reasons or argument, however inconclusive) recourse togeometric concepts or intuitions or principles or understanding, in the attempt to providea satisfactory foundation for real analysis?’, and argues that it was Bolzano. I am grateful toTennant for sharing his typescript with me.

17. Cantor’s idea was first published, with credit, by Heine (1872, p. 173).18. Kant also used that term (e.g., AAXI:208), but, aswewill see, for him it did not refer to infinite

sequences.19. As far as I know, Méray (1835-1911) and Cantor (1845-1918) have never been in contact; in

particular, both as subject and as object Méray is completely absent from Cantor’s known,rich correspondence with the French (Décaillot 2008). Méray states his priority claim on

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ferred to reserve the term ‘nombres’ for whole numbers and rationals (Méray1869, p. 284),20 and considered incommensurable numbers tobe ‘fictions’ (Méray1872, p. 4). The infinite sequences he called ‘variables progressives convergentes’instead. Thus, Méray in effect held a middle position between Kant’s and Can-tor’s. The difference betweenMéray and Cantor is of course by nomeans merelyterminological.21

But even if Kant would have had occasion to consider extending the numberconcept the way Hankel would later suggest,22 he would, as we saw, still haveseen reason to object to the identification of√2 and an appropriate potentiallyinfinite sequence, because of the essential incompleteness of the latter. Now,compare this with Brouwer’s position. Like Kant, Brouwer based his founda-tions of mathematics on an a priori intuition of time.23 Yet, Brouwer acceptsthe modern concept of number and moreover does identify real numbers andcertain potentially infinite sequences:24

We call such an indefinitely proceedable sequence of nested … intervalsa point P or a real number P. We must stress that for us the sequence …itself is the point P. … For us, a point and hence also the points of a set,

p. XXIII of the ‘préface’ to Méray 1894.20. I do not know yet whether a reaction of Méray on Hankel’s work is known.21. For a detailed history of the arithmetization of real numbers, see Boniface 2002 and Petri and

Schappacher 2007.22. One is reminded of the footnote (there, concerning the term ‘analytic’) in section 5 of the

Prolegomena, which begins: ‘It is impossible to prevent that, as knowledge advances furtherand further, certain expressions that have already become classical, dating from the infancyof science, should subsequently be found insufficient and badly fitting ’ (‘Es ist unmöglichzu verhüten, daß, wenn die Erkenntniß nach und nach weiter fortrückt, nicht gewisse schonclassisch gewordne Ausdrücke, die noch von dem Kindheitsalter der Wissenschaft her sind,in der Folge sollten unzureichend und übel anpassend gefunden werden’) (AA IV:276n.)

23. Indeed, in his inaugural lecture ‘Intuitionisme en formalisme’ of 1912, Brouwer presentedhis position as fundamentally Kantian (Brouwer 1913, p. 85). That general qualification isabsent from his later work; in the light of the considerations in the present paper, that seems,conceptually if not historically as well, to be no coincidence.

24. ThatBrouwerhere describes a sequenceofnested intervals, andnotof rationals, is not essentialto the question at hand.

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are always unfinished. (Brouwer 1992, p. 69, original emphasis)25

In what follows, I will leave aside the fact that Brouwer here also includessequences that are constructed not according to a rule but by free choices. EvenBrouwer’s lawlike sequences, such as one for√2, would not themselves be realnumbers for Kant. On the other hand, for Brouwer there is nothing irrationalabout the expression of a magnitude by an incompletable sequence.

Even before we ask whether or not a potentially infinite sequence is a rationalway to express any kind of number, we can ask the ontological question whetherthe concept of such a sequence has constructible instances at all. There are threetheses ofKant’s that, taken together, at first sight seem to lead to apositive answer,along lines very similar to Brouwer:

1. We have an a priori intuition of time;

2. Time is given to us as infinite;

3. ‘Time is in itself a sequence (and the formal condition of all sequences)’(A411/B438).26

Couldn’t a potentially infinite sequence thenbe accepted as an object by labellingmoments in the sequence of time with its elements? We will see, however, thatKant’s understandingof these three theses is such that the answer to this questionis negative.

Lisa Shabel has observed that Kant ‘doesn’t claim that the rule for the approx-imation of an irrational magnitude constitutes a “construction” of any kind’(Shabel 1998, p. 597n. 12); I took that to mean, among other things, that Kantdoes not claim that to generate a potentially infinite sequence of rationals ac-cording to an appropriate rule is to effect the construction of the mathematical

25. ‘Ein derartige unbegrenzte Folge ineinander geschachtelter … Intervalle nennen wir einenPunkt P order eine reelle Zahl P. Wir betonen, dass bei uns die Folge … selbst der Punkt P ist… Bei uns sind ein Punkt und daher auch die Punkte einer Menge immer etwas Werdendes.’

26. ‘Die Zeit ist an sich selbst eine Reihe (und die formale Bedingung aller Reihen).’ In Kemp-Smith’s translation, I have replaced ‘series’ by ‘sequence’.

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concept of an irrational magnitude.27 Here I will defend the stronger thesis thatfor Kant, it would have been impossible to make that particular claim, as in hissystem we have no means to construct the concept of a potentially infinite se-quence. A fortiori, Kant could not have arithmetized irrational quantities byinfinite sequences.

Our considerations must begin with a review of what, for Kant, determineswhether a mathematical concept can be constructed or not.

2 Mathematics within subjective limits

Kant takes the existence of mathematical knowledge as a given (B20). He con-siders philosophy and mathematics two different enterprises that cannot andshould not change one another.28 But they are closely related: mathematics canserve as an instrument in philosophy, and one of the tasks of philosophy is to givean answer to the transcendental question howmathematical knowledge (such aswe indeed have it) is possible.29 In his lectures on logic, Kant emphasizes that thecontent of mathematics as a science is not influenced by the answer to the tran-scendental question. In the explanation what makes mathematical knowledgepossible we cannot find motivations for revisions of mathematics.30

27. In an email, Lisa Shabel has confirmed to me that this indeed is included in her observation; Ithank her for this clarification.

28. Various places inKritik der reinen Vernunft, the Prolegomena; also AA XXIII:201.29. ‘For the possibility ofmathematicsmust itself be demonstrated in transcendental philosophy.”

(‘Denn sogar dieMöglichkeit derMathematikmuß in derTransscendentalphilosophie gezeigtwerden.’) (A733/B761); also A149/B188–189.

30. ‘Welche Bewandtniß es nun aber auch immer hiermit haben möge, so viel ist ausgemacht: injedem Fall bleibt die Logik im Innern ihres Bezirkes, was das Wesentliche betrifft, unverän-dert; und die transscendentale Frage: ob die logischen Sätze noch einer Ableitung aus einemhöhern, absoluten Princip fähig und bedürftig sind, kann auf sie selbst und die Gültigkeitund Evidenz ihrer Gesetze so wenig Einfluß haben, als auf die reineMathematik in Ansehungihres wissenschaftlichen Gehalts die transscendentale Aufgabe hat:Wie sind synthetische Urt-heile a priori in der Mathematik möglich? So wie der Mathematiker als Mathematiker, sokann auch der Logiker als Logiker innerhalb des Bezirks seiner Wissenschaft beim Erklärenund Beweisen seinen Gang ruhig und sicher fortgehen, ohne sich um die außer seiner Sphäreliegende transscendentale Frage des Transscendental-Philosophen und Wissenschaftslehrers

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The starting point for Kant’s transcendental clarification of mathematics ishis dictum ‘Thoughts without content are empty, intuitions without conceptsare blind’ (A51/B75). Mathematical knowledge can only be had if a concept andan intuition of an object are brought together. An intuition is necessary to showthat a concept is related to an object, in other words, that a concept has objectivereality (A155/B194).31 Even instances of analytic judgements count as mathemati-cal knowledge only to the extent that they have been combinedwith appropriateintuitions:

Some few fundamental propositions, presupposed by the geometrician,are, indeed, really analytic, and rest on the principle of contradiction; …And even these propositions, though they are valid according to pureconcepts, are only admitted inmathematics because they can be exhibitedin intuition. (B16–17)32

According to Kant, the only kind of intuition humans have is sensuous in-

bekümmern zu dürfen:Wie reineMathematik oder reine Logik alsWissenschaft möglich sei?’(AA IX:008)

31. For phenomenologists, it is of interest that this is how Kant defines the notion of ‘evidence’:

Whenobjective certainty is intuitive, it is called ‘evidence’ (‘Wenndie obiectiveGewisheitanschauend ist, so heisst sie evidentz.’) (AA XVI:375 (1769? 1770?))

Mathematical certainty is also called evidence, as intuitive knowledge is clearer thandiscursive knowledge. (‘Die mathematische Gewißheit heißt auch Evidenz, weil ein in-tuitives Erkenntniß klärer ist als ein discursives.’) (AA IX:70)

Concepts a priori (in discursive knowledge) can never be a source of intuitive certainty,i.e., evidence, howevermuch the judgementmay otherwise be apodictically certain. (‘AusBegriffen a priori (im diskursiven Erkenntnisse) kann aber niemals anschauende Gewiß-heit, d. i. Evidenz entspringen, so sehr auch sonst dasUrteil apodiktisch gewiß seinmag.’)(A734/B762)

But it is not a term that Kant actually uses often.32. ‘Einige wenige Grundsätze, welche die Geometer voraussetzen, sind zwar wirklich analytisch

und beruhen auf dem Satze des Widerspruchs; … Und doch auch diese selbst, ob sie gleichnach bloßen Begriffe gelten, werden in der Mathematik nur darum zugelassen, weil sie in derAnschauung können dargestellet werden.’

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tuition (A51/B75). This means that we can only have intuitions of objects thatare given to us either in sense perception or in the imagination. Kant denies thathumans can have intuition of (what we would call) abstract objects; we do nothave intellectual intuition. On the other hand, he acknowledges that we do havepurely mathematical knowledge. Kant is able to combine those two views bypointing out that a mathematical concept can be combined with a sensuous in-tuition, namely if the concept is exemplified or instantiated in it.33 In particular,then, on Kant’s conception mathematics is not about sui generis mathematicalobjects, but about possible empirical instantiations of mathematical concepts.34

For example, the concept of the number five is instantiated in an image of fivedots. Moreover, Kant says, when we think of a number (be it small or large) weare not so much thinking of such an image, as of a rule for producing imagesshowing that number of objects (A140/B179). The rule prescribes a series of actsin which an appropriate image will be brought about. Now, the number five willbe equally well instantiated in an image of five dots, strokes, or yet another kindof object. By not stipulating that we use any of these in particular, but merelyrequiring that we be able to consider the things we are adding as in some sensehomogeneous, the rule assumes a generality that accounts for the possibility ofobtaining general knowledge through the acts of producing what is, after all, aparticular image (see also what Kant says on triangles at A713–714/B741–742). Itis here that the inner sense of time comes in. Kant holds that all that we need

33. ‘mathematica per constructionem conceptus secundum intuitionem sensitivam’ (AAXVII:425 (1769? 1773-1775?)); and various other places.

34. ‘mathematics … the object of that science is to be found nowhere except in possible experience’(‘die Mathematik, [die] ihren Gegenstand nirgend anders, als in der möglichen Erfahrunghat’) (A314/B371n.); ‘Consequently, the pure concepts of understanding, even when theyare applied to a priori intuitions, as in mathematics, yield knowledge only in so far as theseintuitions – and therefore indirectly by their means the pure concepts also – can be applied toempirical intuitions’ (‘Folglich verschaffen die reinen Verstandesbegriffe, selbst wenn sie aufAnschauungen a priori (wie in der Mathematik) angewandt werden, nur so fern Erkenntniß,als diese, mithin auch die Verstandesbegriffe vermittelst ihrer auf empirische Anschauungenangewandt werden können’) (B147); ‘[the] mathematician … who likewise deals only withpossible objects of the outer senses’ (‘[der] Mathematiker … der es auch blos mit möglichenGegenständen äußerer Sinne zu thun hat’) (AA XX:418) (1790).

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to be able successively to add units into one image is the inner sense of time(A142–143/B182).35 A rule for producing images that instantiate a number con-cept need therefore not appeal to more than that inner sense. Because of thissufficiency, Kant can say that the foundation of arithmetic (tacitly, as a varietyof human knowledge – see below) is the a priori intuition of time. In the seriesof acts prescribed by a rule, Kant says in his particular idiom, we ‘construct theconcept’ (A713/B741). Such a constructionmay be actually carried out (resultingin, e.g., an actual image of five dots) or, alternatively, be conceived of as an insome appropriate sense ideal possibility (an ideally possible image of one thou-sand clearly distinguishable and surveyable dots) (A140/B179). What matters toKant is not actual construction but ideal constructibility (see also Kant’s replyto Eberhard in this matter: AA VIII:210–212, and the footnote on 191–192). Thisinvites of course a discussion what ‘in principle’ amounts to; for Kant, the ideal-ization involved is constrained by what he takes to be the essential properties ofthe human mind.36

This view on numbers allowedKant to accept as humanly constructiblemath-ematical concepts not only the natural numbers but the rational numbers, too,by taking, to arrive at a particular rational number, whatever part of 1 is appro-priate for unit.37 The concept of such a fractional unit is given intuitive contentgeometrically, by assigning length 1 to a given line segment and then construct-ing geometrically the required part of that segment (for example by the methodof Euclid book VI, proposition 9).

But for Kant, to irrational numbers correspond no humanly constructibleconcepts. As mentioned, Kant held on to the Greek conception of number,which he could readily ground by his particular transcendental account of our

35. Hence, as Kant emphasizes in reflection 6314 (AA XVIII:616 (1790–1791)), for the representa-tion of a number both time and space are necessary, as an image has a spatial character. Seealso 4629 (AA XVII:614) from between 1771 and 1775.

36. In theKritik der Urteilskraft (AA V:254), Kant distinguishes between ‘comprehensive’ and‘progressive apprehension’ (‘comprehensive’ and ‘progressive Auffassung’), but to my mindin both cases what is aimed for is one (ideal) image; here I disagree with vonWolff-Metternich1995, pp. 57–60.

37. Kant does this at, e.g., AA XIV:057 (draft to Rehberg) and AA XI:208 (letter to Rehberg).

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mathematical knowledge:

The concept of magnitude in general can never be explained except bysaying that it is that determination of a thing whereby we are enabled tothink how many times a unit is posited in it. But this how-many-times isbased on successive repetition, and therefore on time and the synthesis ofthe homogeneous in time. (A242/B300)38

In the following elaboration of Kant’s transcendental account, I will refer to anumber of passages in his Reflexionen. Although the Reflexionen in general cancertainly not be granted the same status as Kant’s published work, the specificpassages used below, which are all from 1769 or later, present a coherent view,which in turn is coherent with that presented in the firstKritik.

ForKant, to obtain an image out of amanifold of elements requires a synthesisof the imagination, which necessarily occurs in time. But, as a particularity of thehuman mind, in a finite time span, we can generate a manifold of only finitelymany elements:

Progession. The infinity of the sequence as such is possible, but not the in-finity of the aggregate. The former is an infinite possibility (of additions),the latter an infinite (actual) comprehension. (AAXVII:414, around 1769–1771)39

and, more generally,

What is only given by composition, is for that reason always finite, even

38. ‘DenBegriff derGrößeüberhaupt kannniemand erklären, als etwa so: daß sie dieBestimmungeines Dinges sei, dadurch, wie vielmal Eines in ihm gesetzt ist, gedacht werden kann. Alleindieses Wievielmal gründet sich auf die sukzessive Wiederholung, mithin auf die Zeit und dieSynthesis (des Gleichartigen) in derselben.’

39. ‘Progression. Die Unendlichkeit der Reihe als solche ist möglich, aber nicht die Unendlich-keit des Aggregats. Jenes ist eine unendliche Möglichkeit (der Hinzuthuungen), dieses eineunendliche (wirkliche) Zusammennehmung.’

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though composition can go on infinitely. (AA XVIII:378 no. 5897 around1780–1789?)40

As a consequence, Kant cannot accept any actually infinite totalities as ob-jects of human mathematical knowledge. In particular, it would not be open toKant to accept irrational numbers (and, more generally, real numbers) as actu-ally infinite sums of rational numbers. But he also says that the ground of theimpossibility of infinite composition lies not in the mathematical concept ofinfinity, but in the limits to the capacities of the human mind. Kant does notexclude that minds of a different type can grasp an infinite aggregate as a whole:

When a magnitude is given as a thing in itself, the whole precedes itscomposition, and in that case I cannot conclude from the fact that thisputting together can never be finished and hence its quantitas can neverbe completely known, that such a thing, to the extent that it is an infinitequantum, is impossible. It is only impossible for us to know it completelyaccording to our way of measuring magnitudes, because it is not measur-able. From that it does not follow that a different understanding couldnot know the quantum as such completely without measuring. Similarfor division. (AA XVIII:242–243, no. 5591 (1778–1789))41

(Exactly the same point was already made in the Inaugural Dissertation of 1770(AA II:388 note **).)

Indeed, Kant says explicitly that our impossibility to grasp an infinite mag-nitude as a whole, an impossibility which follows from the dependence of our

40. ‘Was nur durch die composition gegeben wird, ist darum immer endlich, obgleich die com-position ins Unendliche geht.’

41. ‘Wenn eine Größe als ein Ding an sich selbst gegeben ist, so geht das Ganze vor der composi-tion voraus, und da kann ich darum, daß diese zusammensetzung niemals vollendet werdenund also die quantitas derselben niemals ganz erkannt werden kann, nicht schließen, daß einsolches qua unendliche quantum unmöglich sey. Es ist uns nur unmoglich, nach unserer Artgroßen zu messen es gantz zu erkennen, weil es unermeßlich ist. Daraus folgt nicht, daß nichtein anderer Verstand ohneMessen das quantum als ein solches Ganz erkennen könne. Ebensomit der Teilung.’ Also e.g., AA XVIII:379 no. 5903.

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grasp of magnitudes on time, is not objective, but subjective:

In the infinite, the difficulty is to reconcile the totality with the impossi-bility of a complete synthesis. Therefore the difficulty is subjective. Onthe other hand, the potential infinite (infinity of potential coordination)is very well understandable, but without totality. (AA XVII:452 no. 4195.1769–1770?)42

and

How the conflict with subjective conditions or their presupposition mir-rors the truth of the objective conditions and forces itself upon [unter-schiebe] the latter. For example, a mathematical infinite is possible, as itdoes not conflict with the rules of the intellect [der Einsicht]; it is impos-sible, as it conflicts with the conditions of comprehension. (AA XVIII:1351776–1778?)43

Kant’s answer to the question what these subjective limits are for us is ‘thatwhich can be represented a priori in intuition, that is, space and time and changein time’. (AA XVII:701 (around 1775–1777))44 I take it, then, that Kant’s remarkin the quotation above from A242/B300 that the explanation of the notion ofmagnitude must depend on the notion of successive repetition and hence ontime is limited to the specific context of human mathematical cognition, andthat the same also holds for his statement at A142–143/B182 that ‘Number istherefore simply the unity of the synthesis of the manifold of a homogeneous

42. ‘Im Unendlichen ist die Schwierigkeit, die totalitaet mit der unmöglichkeit einer synthesiscompletae zu vereinbaren. folglich ist die Schwierigkeit subiectiv. Dagegen ist das potentialiterinfinitum (infinitum coordinationis potentialis) sehr wohl begreiflich, aber ohne totalitaet.’

43. ‘Wie derWiederstreit der subiectiven Bedingungen oder ihre Voraussetzung dieWahrheit derobiectiven nachahme und unterschiebe. e. g. Ein Mathematisch unendliches ist möglich, weiles den regeln der Einsicht nicht wiederstreitet; es ist unmöglich, weil es den Bedingungen dercomprehension wiederstreitet.’

44. ‘Welches sind die Grenzen der mathematischen Erkenntnis? Das, was a priori in der Anschau-ung kann vorgestellt werden, also Raum und Zeit und Veranderung in der Zeit.’

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intuition in general, a unity due tomy generating time itself in the apprehensionof the intuition’.45 I will now turn to the question what Kant’s conception ofmathematics within the subjective limits proper to us means for his view on realnumbers.46

3 Kant’s discussion with Rehberg

Rehberg’s primary concernwhenwriting toKant in 1790was not the ontologicalstatus of real numbers, but the issue whether the intuition of time is really acondition of the possibility of mathematics for us. For our present purpose, themain interest of Rehberg’s letter lies in two specific questions that are raised init:

45. ‘Also ist die Zahl nichts anderes, als die Einheit der Synthesis des Mannigfaltigen einer gleich-artigen Anschauung überhaupt, dadurch, daß ich die Zeit selbst in der Apprehension derAnschauung erzeuge.’

46. Maimon, in his Versuch über die Transscendentalphilosophie, also emphasizes the dependencyon subjective conditions. Describing the division of a line segment into parts, he writes:

In case the parts are infinite [in number], then this division is, for a finite being, impossi-ble, not, however, in itself. (Maimon 1790, p. 375) (‘Sind also die Theile unendlich, so istdiese Theilung, in Beziehung auf ein endliches Wesen, unmöglich, nicht aber an sich.’ )

And, on infinite numbers:

An absolute understanding, on the other hand, thinks the concept of an infinite num-ber without invoking a temporal sequence, all at once. Therefore, that which for theunderstanding [i.e., the human understanding] is, in accordance with its limitations, amere idea, is, with respect to its absolute existence, a true object. (Maimon 1790, p. 228)(‘Bei einem absolutenVerstande hingegen, wird der Begrif einer unendlichen Zahl, ohneZeitfolge, auf einmal, gedacht. Daher ist das was der Verstand [i.e., the human under-standing] seiner Einschränkungnach, als bloße Idee betrachtet, seiner absolutenExistenznach ein reelles Objekt.’)

It seems, then, that Maimon explicitly leaves open the possibility that infinite minds couldadmit into arithmetic not only whole and rational numbers, but also real numbers, as actuallyinfinite sums of fractions. The human mind, however, cannot do this.

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1. What are the conditions of the possibility of knowing that √2 is irra-tional? Rehberg disputes Kant’s claim at A149/B188–189 in theKritik derreinen Vernunft that ‘mathematical principles … are derived solely fromintuition, not from the pure cocnept of understanding’.47 WhileRehbergagrees for the case of geometry, he disagrees for the case of arithmetic andalgebra, and claims that in those domains the a priori intuitions of timeand space are not necessary to obtain knowledge, but only the conceptsthemselves (AA XI:205–206). In his later comments, Rehberg calls thecorresponding kind of intuition ‘pure intellectual intuition ’ (‘reine An-schauung des Verstandes’) (Rehberg 1828, p. 57).

2. ‘Why is our understanding, which produces numbers spontaneously, un-able to think numbers corresponding to√2?’ (AAXI:206)48 It is not clearwhat Rehberg exactly means by ‘thinking a number’,49 but the very factthat Rehberg, who must have known the infinite series, raises this ques-tion, suggests that generating an infinite series is not an example.50

47. ‘[D]ie mathematischen Grundsätze … [sind] nur aus der Anschauung, aber nicht aus demreinen Verstandesbegriffe gezogen’.

48. ‘Warumkann er [i.e., , der Verstand], der Zahlenwillkührlich hervorbringt keine√√2Zahlendenken?’ From Rehberg’s letter and his later elaboration of his view (Rehberg 1828, p. 56), itis clear that by ‘willkürlich’ he does not mean ‘subject to no condition at all’. While he claims,against Kant, that it is a spontaneity that is unconstrained by the forms of time and space, healso thinks it is subject to constraints of a different kind (see footnote 50 below), and takes theimpossibility, as he sees it, to think√2 in numbers as a proof of that fact.

49. Longuenesse claims that Rehberg means by it ‘thinking in multiples or fractions of the unit,that is, in rational numbers’ (Longuenesse 1998, p. 262n. 38). (AlsoDietrich reads him thatway(Dietrich 1916, p. 118).) I do not find evidence for this in Rehberg’s letter or his later comments.In effect, on that reading Rehberg is asking why the understanding cannot think an irrationalnumber as a rational one. I read Rehberg differently; see the next footnote. (Of course, whenRehberg writes, ‘Es heißt zwar p. 182 der Critik, daß die Zahl eine successive Addition sey’(AA XI:205), this formulation is neutral as to whether he agrees.)

50. Rehberg’s own suggestion for an answer is that the ground of this impossibility lies in the tran-scendental faculty of the imagination and its connection to the understanding (AA XI:206),which he thinks has a property that limits our capacity of generating numbers in such a waythat thinking [a quantum] in numbers for us is limited to ‘discretely generated magnitudes’(‘discretive erzeugten Größen’) (Rehberg 1828, pp. 57,59); see also Parsons 1984, p. 111. In

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In his reply to Rehberg, Kant argues that time is, after all, involved in ourcoming to know the irrationality of √2, as follows.51 From the mere conceptof a given natural number it cannot be seen whether its square root is rationalor irrational. To determine this, Kant appeals to the following theorem: if thesquare root of a natural number n is not itself a natural number, then it is nota rational number either (AA XI:209).52 We can only find out whether n is thesquare of a natural number by testing. The test proceeds by constructing thenatural numbers from 1 onward until the square is equal to or greater than n.53But constructing numbers involves the intuition of time. And, although Kant

his letter he qualifies the nature of this faculty as ‘transcending all human capacities of in-vestigation’‘ (alles menschliche Untersuchungsvermögen übersteigend’) (AA XI:206), butnevertheless goes on to suggest the possibility of a ‘transcendental system of algebra’ (‘trans-scendentales System der Algebra’), which would serve to determine a priori, on the basis ofprinciples, which equations we can solve and how. In one of the drafts for his reply, Kantsays he can answer Rehberg’s question ‘without having to look into the first grounds of thepossibility of a science of numbers’ (‘ohne auf die ersten Gründe der Moglichkeit einer Zahl-wissenschaft zurüksehen zu dürfen’) (AA XIV:55–56), but it is interesting that, decades earlier,he himself in a note had remarked: ‘Philosophical insight into geometrical and arithmeticalproblems would be excellent. It would open the way to an art of discovery. But it is very diffi-cult. ’ (‘Ein philosophisch Erkentniß der geometrischen undArithmetischenAufgabenwürdevortreflich seyn. sie würde den Weg zur Erfindungskunst bahnen. aber sie ist sehr schweer.’)(AA XVI:55 (1752–W. S. 1755/56))

51. Given Kant’s remarks quoted at the end of Sect. 2, I disagree with Friedman’s claim that forKant, ‘the fact of the irrationality of√2, which is presumably a fact of pure arithmetic, is itselfbased on successive enumeration andhence on time’ (Friedman 1992, p. 116, original emphasis).What depends on time is rather the possibility for humans to come to know that fact. See alsoKant’s letter to Schultz of November 25, 1788 (AA X:556–557) and Parsons’ comments on it(Parsons 1984, pp. 116–117).

52. This is known as Theaetetus’ Theorem, although Plato’s dialogue to which it owes its namegives no proof; for the ancient history of the theorem and its proofs, see Mazur 2007. Kant(who does not call the theoremby that name)maywell have seen it, with a proof, in Sect. 137 ofJohann Segner’sAnfangsgründe der Arithmetik (Segner 1764) towhich he refers, in a differentcontext, at B15. The method to extract the square root of larger numbers that Kant refers toat AA XI:209 corresponds to the method given by Segner in Sect. 136. (The same material isalso present in Michael Stifel’sArithmetica Integra (Stifel 1544) of which Kant owned a copy(Warda 1922, p. 40).)

53. Note that the procedure to extract roots in effect starts with the same test.

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does not remark on this in the letter, more generally, for him any algebraicmeansof establishing the irrationality of √2 could be said to depend on the a prioriintuition of time, as for him it is characteristic of an algebraic proof as such thatit ‘exhibits all the procedure through which magnitude is generated and alteredin accordance with certain rules in intuition’ (A717/B745).54 ‘Step by step’, asFriedman comments on that statement (Friedman 1992, p. 120n. 42).

It is in reply to Rehberg’s second question, why the understanding cannotthink√2 in numbers, that Kant rejects, as we have seen, the identification of thatsquare root with a certain potentialy infinite sequence, because of the essentialincompletability of the latter. In the following, I attempt to reconstruct theground on which for Kant this incompletability is objectionable.

4 Infinite sequences as concepts and as objects

One difference between a potentially infinite sequence and an image is that theparts of an image all exist simultaneously,whereas the parts (elements) of a poten-tially infinite sequence do not.55 In an image, the elements of the sequence thatare not yet there obviously cannot be shown. Moreover, the fact that there arefurther elements yet to come, which is part of the concept of potentially infinitesequence, cannot itself be intuitively represented in the image.56 This is because

54. ‘so stellet sie alle Behandlung, die durch die Größe erzeugt und verändert wird, nach gewissenallgemeinen Regeln in der Anschauung dar’.

55. Compare AA XVII:397 no. 4046 (1769? 1771?): ‘The omnitudo collectiva in One or totalityrests on the positione simultanea. From the multitudine distributiva I can conclude to theunitatem collectivam, but not from the omnitudine, because the progression is infinite andnot complete.’ (‘Die omnitudo collectiva in Einem oder totalitaet beruhet auf der positionesimultanea. Aus der multitudine distributiva kan ich auf die unitatem collectivam schließen,aber nicht aus der omnitudine, weil die Progression unendlich ist und nicht complet.’); alsoAAXVII:700 (around 1775–1777): ‘The infinite of continuation or of collection.The infinitelysmall of composition or decomposition. Where the former is the condition, the latter doesnot occur.” (‘Unendlich der Fortsetzung oder der Zusammennehmung. unendlich klein dercomposition oder decomposition. Wo das erstere die Bedingung ist, findet das letztere nichtstatt.’

56. Note that ideal, adequate givenness of a potentially infinite sequence does not consist in itsbeing given as an actually infinite sequence (for that would contradict the essence of the object

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forKant, there is nothing to the sequence that can be given intuitively, and hencesynthesized, but the elements constructed so far themselves.57 (When we write0.333 . . ., we understand what the three dots stand for, but the concept theyinstantiate is not that of infinity but of the number 3.) The understanding givesform to our sensuous intuitions by combining them in certain ways, but theseforms are not themselves given to us in their own kind of intuition. In Husserlthere is categorial intuition, but not in Kant.58 Rather, Kant characterizes thehuman understanding as one

whose whole power consists in thought, consists, that is, in the actwhereby it brings the synthesis of a manifold, given to it from elsewherein intuition, to the unity of apperception – a faculty, therefore, which byitself knows nothing whatsoever, but merely combines and arranges thematerial of knowledge, that is, the intuition, which must be given to it bythe object. (B145)59

qua potentially infinite), but in the givenness of the whole finite initial segment generatedso far, however large the number of its elements may be, together with the open horizonthat indicates the ever present possibility to construct further elements of the sequence. Theabsence of such further elements from an intuition of the sequence at a given moment doesnotmake it an inadequate intuition, because they do not yet even exist. In contrast, the reasonwhy our intuition of a physical object at a given moment is necessarily inadequate is preciselythat, as a matter of three-dimensional geometry, any concrete view of it hides parts that do atthat moment exist.

57. The order relation is represented by the relation between left and right, but that alreadyrequires an act of the understanding: do we take the order in a sequence to be from left toright, or from right to left?

58. ‘It is true that in Kant’s thought the categorial (logical) functions play a great role; but henever arrives at the fundamental extension of the concepts of perception and intuition overthe categorial realm’ (‘In Kants Denken spielen zwar die kategorialen (logischen) Functio-nen eine große Rolle; aber er gelangt nicht zu der fundamentalen Erweiterung der BegriffeWahrnehmung und Anschauung über das kategoriale Gebiet.’) (Husserl 1984, p. 732).

59. ‘dessen ganzes Vermögen im Denken besteht, d. i. in der Handlung, die Synthesis des Man-nigfaltigen, welches ihm anderweitig in der Anschauung gegeben worden, zur Einheit derApperception zu bringen, der also für sich gar nichts erkennt, sondern nur den Stoff zumErkenntniß, die Anschauung, die ihm durchs Object gegeben werden muß, verbindet undordnet’. See also A51/B75, B138–139, A147/B186, B302–303n., A289/B345,Prolegomena sections

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We can therefore represent a potentially infinite sequence as a concept, and in-deed use the concept to construct ever longer finite sequences, but we can neverwholly instantiate that concept itself in an intuition.Hence, forKant the conceptof such a sequence is not mathematically constructible.

Note that the impossibility of a potential infinite sequence as a constructiblemathematical concept has its ground in the requirement of an image rather thanin a property of our capacity of synthesis. For it is the requirement of an imagethat imposes a condition of completeness (i.e., , the simultaneous presence ofall its parts).60 This is also why for Kant it is irrational to try to arrive at a rep-resentation of a quantum by generating a potentially infinite sequence. On theother hand, Kant acknowledges that in principle the acts of synthesis can alwaysbe continued:

22, 39 and 57.60. According to Kant, in pure mathematics all questions have a definite answer (or else the

senselessness of the question can be demonstrated), and the same holds for transcendentalphilosophy and pure ethics (A476/B504ff.); see for discussion Posy 1984, pp. 127–128. Thegeneral reason Kant gives for this is that in these purely rational sciences, ‘the answer mustissue from the same sources from which the question proceeds’ (‘die Antwort aus densel-ben Quellen entspringen muß, daraus die Frage entspringt’) (A476/B504). It seems to methat, when the details of this answer are spelled out for the case of pure mathematics, thecondition of completeness that is imposed by Kant’s requirement of an image must enterinto the explanation. For intuitionistic mathematics is equally wholly concerned with spon-taneous constructions in a priori intuition – where Kant speaks of questions raised by purereason as concerned with its ‘inner constitution’ (innere Einrichtung) (A695/B723), Brouwercalls mathematics ‘inner architecture’ (Brouwer 1949, p. 1249). But in intuitionism, the mostwe can justify in general is the weaker claim that there are no unanswerable questions, as¬¬(p ∨ ¬p) is demonstrable while p ∨ ¬p is not. For example, consider a potentially infi-nite lawless sequence of natural numbers α (which, as follows from the considerations in thepresent paper, for Kant would not be a mathematically constructible concept). We cannot,in general, show that ∃n(α(n) = 0)∨ ¬∃n(α(n) = 0), due to the open-endedness of such asequence. We can show ∃n(α(n) = 0) as soon as we have indeed chosen 0 in the sequence,but we are never obliged to make that choice. On the other hand, we can at any time show¬¬(∃n(α(n) = 0) ∨ ¬∃n(α(n) = 0)) (which also shows that the original question is notsenseless). Intuitionism, however, accepts Kant’s claim for questions that ask whether a givenconstruction of finite character is possible in a given finite system; e.g., Brouwer 1949, p. 1245.

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The infinity of synthesis in a sequence [is], as in a progression, only po-tential. (AA XVIII:277)61

and, to repeat an earlier quotation,

What is only given by composition, is for that reason always finite, eventhough composition can go on infinitely.’62

Ofcourse,wemay create an imageof a finite sequence to construct the conceptof an initial segment of a potentially infinite sequence, but the potentially infi-nite sequence is not thereby itself given in intuition. The difference is both philo-sophically and mathematically important: the collection of all finite sequencesis denumerable, the collection of all potentially infinite sequences is not.

We also can associate to the concept of the potentially infinite sequence aschema, as a method to construct in intuition ever longer initial segments. In-deed, in his letter to Rehberg, Kant says that√2 ‘is actually no number, but onlya determination of magnitude bymeans of a rule of enumeration’, and he seemsto hold this for real numbers more generally.63 But as Kant emphasizes, in a dif-ferent text and for a different reason, a schema is not itself an image (A142/B181).So Kant is not only saying that√2 is no number (in his sense), but that it is noproper object. Note also that the rule is, in one sense, given in intuition whenwritten down. But that is not the sense needed here: the written rule is a finiteobject, whereas what is under discussion here is the intuition of a potentiallyinfinite sequence.64

61. ‘Die Unendlichkeit der Synthesis in einer Reihe [ist] wie im progressu blos potential.’62. AA XVIII:378 no. 5897 around 1780–1789?: ‘Was nur durch die composition gegeben wird,

ist darum immer endlich, obgleich die composition ins Unendliche geht.’63. ‘eine Irrationalzahl … ist … wirklich keine Zahl, sondern nur eine Großenbestimmung durch

eineRegel des Zählens’ (AAXIV:57) Compare in one of the drafts: ‘a square root…, but alwayssuch a one that is itself no number, but only the rule of approximation to it, however far onedemands’ (‘eine Qvadratwurzel …, allenfalls eine solche, die selbst keine Zahl, sondern nurdie Regel der Annäherung zu derselben, wie weit man es verlangt’) (AA XI:210).

64. Compare on this point also Wittgenstein: ‘ “We know the infinity from the description.”Well, then only this description exists and nothing else.’ (‘ “Wir kennen die Unendlichkeit

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Kant’s acknowledgement that composition can go on infinitely certainly in-volves a knowledge that time is in some sense infinite, as all composition takesplace in, and hence presupposes, time. What, then, of the suggestion (above,p. 8) that Kant’s theses of time as infinite, given, and sequential, could provide abasis for the construction of infinite sequences?

Kant says that time, in its original representation, is not a concept, but is givento us, and as unlimited at that:

5. The infinitude of time signifies nothing more than that every determi-nate magnitude of time is possible only through limitations of one singletime that underlies it. The original representation, time, must thereforebe given as unlimited. But when an object is so given that its parts, andevery quantity of it, can be determinately represented only through limi-tation, the whole representation cannot be given through concepts, sincethey contain only partial representations; on the contrary, such conceptsmust themselves rest on immediate intuition. (B47–48)65

ButKant denies thatwe can represent time itself in amode of intuition properto it, and repeatedly says that time itself cannot be perceived, e.g.:66

For time is not viewed as that wherein experience immediately determines

aus der Beschreibung.” Nun, dann gibt es eben nur diese Beschreibung und nichts sonst.’)(Wittgenstein 1964, p. 155).

65. ‘5) Die Unendlichkeit der Zeit bedeutet nichts weiter, als daß alle bestimmte Größe der Zeitnur durch Einschränkungen einer einigen zum Grunde liegenden Zeit möglich sei. Dahermuß die ursprüngliche Vorstellung Zeit als uneingeschränkt gegeben sein. Wovon aber dieTeile selbst, und jede Größe eines Gegenstandes, nur durch Einschränkung bestimmt vorge-stellt werden können, da muß die ganze Vorstellung nicht durch Begriffe gegeben sein (denndie enthalten nur Teilvorstellungen), sondern es muß ihnen unmittelbare Anschauung zumGrunde liegen.’

66. Here also, Brouwer andHusserl disagree with Kant; e.g., Brouwer (1907, pp. 104–105), claimsthat the one-dimensional temporal intuitive continuum is given as anobjectwithout requiringthe givenness of any other object; for Husserl, see Husserl 1928, in particular pp. 436–437 and471–473.

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position for every existence. Such determination is impossible, inasmuchas absolute67 time is not an object of perception with which appearancescould be confronted. (A215/B262)68

According to Kant, we can represent time as an object only indirectly, by analogy(A33/B50), ‘under the image of a line, in so far as we draw it’ (‘unter dem Bildeeiner Linie, so fern wir sie ziehen’) (B156).69 As soon as we conceptualize time,that is, come to think of it as an object to which concepts apply, then it has tobe represented by a construction in space.70 Indeed, for Kant the intuitivenessof our representation of time is concluded to from the possibility to represent itspatially, and we derive all properties of time not from a direct representation ofit, but from the line (A33/B50) (except that the reference to the act of drawing isessential for the representation of succession (B154–155)).

A consequence for Kant’s view is that the intrinsic possibilities and limita-tions of spatial representation also condition our representation of time as anobject. In the Transcendental Aesthetic, Kant argues that space is given to usas infinite (B39–40). An elucidation is given in a later manuscript, ‘Über Käst-ners Abhandlungen’ of 1790.71 Kant there distinguishes between mathematical

67. [By ‘absolute’, I take Kant here to mean ‘not in relation to any objects whose appearances aretemporally determined’, in analogy to his explanation of the term ‘absolute space’ in the noteat A429/B457.]

68. ‘die Zeit wird nicht als dasjenige angesehen, worin die Erfahrung unmittelbar jedem Daseinseine Stelle bestimmte, welches unmöglich ist, weil die absolute Zeit kein Gegenstand derWahrnehmung ist’. Also A32–33/B49, A37/B54, B219, B225, B226, B233, B257.

69. Following Böhme (1974, p. 272), I take it that Kant is referring not to time as such but to timein this relation to space and movement when he writes that ‘The pure image … of all objectsof the senses in general is time’ (‘Das reine Bild … aller Gegenstände der Sinne aber überhaupt,die Zeit’) (A142/B181–182).

70. AA XIV:55 (1790): ‘But without space, time itself would not be represented as a magnitudeand this concept would have no object at all.’ (‘Aber ohne Raum würde Zeit selbst nicht alsGröße vorgestellt werden und überhaupt diese Begriff keine Gegenstand haben.’)

71. AA XX:410–423, in particular 417ff. Written for, and indeed used by, Johannes Schultz; seethe latter’s ‘Rezension von Johann August Eberhard, Philosophisches Magazin’ (Schultz1790), and Kant’s letters to Schultz of Summer 1790: AA XI:183, AA XI:184, AA XI:200, AAXI:200–201.

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infinity and metaphysical infinity. It is the latter that according to Kant is ‘anactual (but only metaphysically real) infinity’.72 It is actual because it is presentin all of our experiences, and Kant therefore says this infinity is given to us. It isalso metaphysical, because by that qualification Kant means that it pertains tothe subjective forms of our sensibility. (The more usual notion of metaphysicsKant refers to as ‘dogmatic metaphysics’.) At the same time, Kant repeats thepoint he had made, in somewhat different words, in the Transcendental Aes-thetic (B39) that actual, metaphysical space cannot be brought under a conceptthat we would be capable of constructing. In fact, metaphysical, actual infinityis the precondition for the potential infinity of our mathematical constructions.It is the former that guarantees the presence of indeterminate space in whichmathematicians construct determinate parts.73

As any such constructed determinate part will be finite, we can represent ina determinate way only finite segments of time in spatial intuition. When werepresent such a finite segment of time by a finite line, the part of time thatis yet to come, the future, is represented in an indeterminate way by the partof metaphysical, given space into which we have not yet extended the line butcan do so if we wish.74 But as according to Kant metaphysical space as suchis unconceptualizable, the finite line we have drawn and metaphysical, givenspace do not together make up an image in which the concept of a potentiallyinfinite segment of time is constructed. Metaphysical space is not an image orpart thereof, but a condition of possibility for images (see also footnote 69). This

72. That concise phrase occurs in a longer passage that Kant deleted; but the content of the pas-sage agrees with themain text (in particular pp. 420–421). The sentence containing this phraseruns: ‘Denn daß man eine Linie ins Unendliche fortziehen oder Ebenen so weit man will auseinander rücken kan diese potentiale Unendlichkeit welche der Mathematiker allein seinenRaumesbestimmungen zum Grunde zu legen nöthig hat setzt jene actuelle (aber nur meta-physisch wirkliche) Unendlichkeit voraus und ist nur unter dieser Voraussetzung möglich.’(AA XX:418).

73. As the Transcendental Aesthetic is concerned with metaphysical infinity, not mathematicalinfinity, it gives necessary, but not sufficient conditions for mathematical cognition. Theseneed to be completed by the Axioms of Intuition. See for a detailed discussion of this pointSutherland 2005.

74. See on this point Michel 2003, p. 112.

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means that we cannot represent time in intuition as a potentially infinite object.It follows that, although there is for Kant a specific sense in which time is givento us and is given to us as unlimited, this does not provide us with a basis for theconstruction of the concept of a potentially infinite sequence.

5 Concluding remark

The above arguments are general: for Kant the concept of no potentially infinitesequence whatsoever can be constructed by us, be it in a mathematical contextor not. An incompletable process, even when fully specified, can never result inone, finished image.75 In the case of the natural numbers, this means that Kant’sposition allows him to construct every one of them, one after the other, butnot the potentially infinite sequence of them. It also means that Kant’s positiondoes not allow him to identify real numbers with potentially infinite sequences.(Likewise, any other explicit construction of a real number as an object out ofinfinitely many elements, such as a Dedekind cut, is impossible.) This changeswhen one recognizes what Husserl called ‘categorial intuition’, and accepts thatthe flow of time, together with its structuring moments of retentions and pro-tentions, is given in an intuition proper to it; for this opens the possibility ofapplying categorial intuition to the flowof time, and then on that basis constructpotentially infinite sequences as objects in intuition, as Brouwer did. That leadsto a far richer mathematics.76

Acknowledgements. Earlier versions were presented at CUNY,NewYork,Novem-ber 6, 2008; at REHSEIS, Paris, January 16, 2009; at the Oskar Becker Tagung,Bad Neuenahr/Ahrweiler, February 6, 2009; at IHPST, Paris, March 23, 2009;

75. In this sense, for Kant potentially infinite sequences would seem to be evenmore problematicthan actually infinite ones; the latter might still be representable in an image by other mindsthan ours.

76. This is not to suggest that Husserl actually influenced Brouwer; rather, in my view, the ideasthat Brouwer independently developed are best understood in the framework that Husserlprovides. See van Atten 2007 for a phenomenological analysis of Brouwer’s choice sequences.

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at Philosophy and Foundations of Mathematics: Epistemological and Ontolog-ical Aspects (a conference dedicated to Per Martin-Löf on the occasion of hisretirement), Uppsala, May 7, 2009; at the meeting of the Société des études kan-tiennes en langue française, Lyon, September 8, 2009; at the joint philosophy-mathematics seminar (CEPERC/UFRAM) in Marseille, March 10, 2010; and atthe logical-philosophical seminar at Charles University, Prague, March 28, 2011.I thank the audiences for their questions and comments, and also Carl Posy,Ofra Rechter, Lisa Shabel, Pirmin Stekeler-Weithofer, Neil Tennant, RobertTragesser, and an anonymous referee.

References

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A Kritik der reinen Vernunft. First edition.Hartknoch,Riga, 1781. Editionused: W. Weischedel (ed.), Suhrkamp, Frankfurt, 1974.

AA Gesammelte Schriften. Vols. I–XXIX. Hrsg. von der Königlich-Preussischen Akademie der Wissenschaften zu Berlin, 1902–.

B Kritik de reinen Vernunft. Second edition. Hartknoch, Riga, 1787. Edi-tion used: W. Weischedel (ed.), Suhrkamp, Frankfurt, 1974.

English translations of AA are my own; those of A and B are taken fromN.Kemp Smith’s translation Immanuel Kant’s Critique of Pure Reason, St.Mar-tin’s Press, New York, 1965.

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